3.8 On discrete probability approximations
3.8.2 Binomial discrete probability approximation
In this section we consider a simple binomial model for risky growth evolution. To obtain some analytic insights for the model in section 3.8.1, let us define the risky growthG∗ as the ratio of future time step stock price and to the previous time step stock price i.e. G∗ = Sk+1
Sk . In
this section we consider a simple binomial model for risky growth evolution. To obtain some analytic insights for the model in section 3.8.1, let us define the risky growthG∗ as the ratio of future time step stock price and to the previous time step stock price i.e. G∗ = Sk+1
Sk . To
obtain some analytic intuition assume that the risky growthG∗evolves binomially. It is (1+g)
with a probability of p and (1−g) with a probability of (1− p). The dynamic programming problem discussed in the previous section still holds and so each stage involves a non-linear optimization problem. To find the risky boundaries at the previous stage stage we setAk = 0
or Ak = 1 in the beginning of the period to force the buy or selling of the risky asset and
hence determine the buy and sell-side boundaries. To get exact analytic formulas, however, we linearize with respect toλ,µandV. This approximately holds valid for some range of values for the transaction cost and risk aversion parameters. This helps us determine (φ−, φ+) so that we could get some analytic insights into the risky boundaries.
Let us see how the approximation works for the last stage of the portfolio problem. In the analysis we will assumeV ∈[0,1]
Start with the buy-side boundary approximation. If A > 0 denotes the buy-side boundary then it satisfies the following non-linear equation as a result of optimization on control in the dynamic programming equation given in Section (3.8.1) :
p(s+λsA−sA+A(1+g))V +(1− p)(s−λsA−sA+A(1−g))V = 0 (3.51) where sis the growth rate over the interval for risk-free asset.
However, in the above observe the co-efficients of A. If we assumek 1+g
s −(1+λ) kand
k 1−sg −(1+λ) kare small14 for some small transaction costs then we can linearize to get the
buy-sideboundary: φ−= inf(sup(a+bλ,0),1) (3.52) where a = s(−2gp+g+ s−1) (V −1)(g2−2g(2p−1)(s−1)+(s−1)2) (3.53) b = s 2(g2(8p2−8p+1)−2g(2p−1)(s−1)+(s−1)2) (V−1)(g2−2g(2p−1)(s−1)+(s−1)2) (3.54)
Similarly consider with the sell-side approximation.IfA< 1 denotes the sell-side boundary then it satisfies the following non-linear equation as a result of optimization on control in the dynamic programming equation given in Section (3.8.1) :
14 1+g
1+s −(1+λ)= g−s
3.8. On discrete probability approximations 37
p(s+A(1+g−s)+µ(A−1)s)V +(1− p)(s+A(1−g−s)+µ(A−1)s)V = 0 (3.55) where sis the growth rate over the interval for risk-free asset.
However, in the above observe the co-efficients of A. If we assumek 1+sg −(1−µ) kand k 1−g
s −(1−µ) k are small for some small transaction costs then we can linearize to get the
sell-sideboundary:
φ+=sup(inf(a+dλ,1),0) (3.56)
whereais defined earlier and
d = s(g
3(2p−1)+g2(8p2−8p−2s+3)+g(2p−1)(s2−4s+3)+(s−1)2)
(V −1)(g2−2g(2p−1)(s−1)+(s−1)2) (3.57)
Provided E[G∗]> s−1 we will showa> 0 andb< 0 for some range of parameters which agrees with our intuition.
Holding all else constant increasing V increasesa,band d while decreasing V decreases the same parameters.
AssumingE[G∗]> s−1 results ing(2p−1)> s−1. Further we assume 0< p<1,1< s<2 and 0< g<1.
Proposition 3.8.1 In the portfolio model (3.43)-(3.49) a =c> 0:
Proof Examine a. To prove a > 0 we need to find the signs of terms in the numerator and
denominator. Now −(2p −1) > −1 implies −2g(2p− 1)(s −1) > −2g(s− 1) implies g2 −
2g(2p−1)(s−1)+(s−1)2 >(g−(s−1))2> 0.
Since g(−2p+1)+(s−1)<0as described earlier we can state:
a= c>0
Proposition 3.8.2 In the portfolio model (3.43)-(3.49) b <0if p∈[p∗,1− p∗]:
Proof Let us examine b. To prove b < 0 we need to find the signs of terms in the numerator and denominator. Now g(2p−1)> s−1implies−2g(2p−1)(s−1)<−2(s−1)2which implies
−2g(2p−1)(s−1)+(s−1)2< −(s−1)2 <0. So if8p2−8p+1< 0then b< 0.
In short: b < 0 if p∗ < p < 1− p∗,E[G∗] > s−1 where p∗ is obtained by solving the
quadratic equation8p2−8p+1< 0.
The signs ofa,cmatch the financial intuition behind the structure of no-transaction bound- aries and thus provides support to our analytics. Note that it isn’t always true that d < 0 as this statement holds only for a particular set of parameters. We will illustrate numerically the dependence of no-transaction boundaries on transaction cost at different stages of dynamic problem for some choice of parameters in Figure 3.7.
38Chapter3. Introduction to discrete probability approximation and sketch of modeling approach
This was a short introduction to discrete probability approximations which are really mo- ment and cross-moment based approximations for risky growth. But what are the best mo- ments to match? Are these the moments of risky growths or moments for some function of risky growth? The nature of the value function provides the key to answer these. Most often if the risky growth appears linearly in the value function the best way is to choose moment based approximation for the risky growth itself. The thesis will discuss in detail the methodology and provide some applications.